Macroscopic Approximations to Networks with Applications to Performance Analysis

研究了获取通信网络宏观近似的技术研究与应用。这些近似是确定性的，用常微分方程描述，是随机的（扩散），用随机微分方程描述。前者给出了时间相关行为的近似，而后者量化了统计偏差。该方法是弱收敛和鞅方法的结合。所提出的方法允许对服务时间不仅具有指数分布而且具有一般服务时间分布的网络进行研究。传输速率可以依赖于缓冲区占用，这在拥塞控制应用中很重要。由于不存在灵活的分析工具，无法估计这种（非产品形式）网络的队列长度和其他性能指标，因此这些近似的重要性变得明显。通过某些限制程序产生的近似模型将用于进一步研究稳定性问题、稳定时间和收敛到稳态的速率。这些方法和估计将在数学上建立，并通过数值和模拟进行验证。

The investigation and application of techniques for obtaining macroscopic approximations for communication networks is studied. These approximations are deterministic, described by ordinary differential equations, and stochastic (diffusion), described by stochastic differential equations. The former yield approximations for the time dependent behavior, while the latter quantify the statistical deviations. The techniques consist of a combination of weak convergence and martingale methods. The proposed methods allow for the investigation of networks with service times having not only exponential but also general service time distributions. The transmission rates can be taken to depend on the buffer occupancies, which is important in congestion control applications. The importance of those approximations becomes clear given the non-existence of flexible analytical tools that can give estimates of queue lengths and other performance measures for such (non-product-form) networks. The approximate models, arising via certain limiting procedures, will be used to further study stability issues, settling times and rates of convergence to the steady state. The methods and estimates will be established mathematically and also validated numerically and by simulations.